Perfect Figures

The Lore of Numbers and How We Learned to Count

Bunny Crumpacker

Thomas Dunne Books

Perfect Figures
1
ONE IS ALL
One is the beginning, the single starting place. It's the universe at the big bang: There was that enormous event, that unthinkable noise, and suddenly, in a fraction of a second, whatever there was--had it been One?--had shattered. It became a billion million stars, galaxies after galaxies of stars, stars with planets and moons and meteors and asteroids, each one containing everything again, atoms and molecules, charm and quark, and each thing--each atom, each galaxy--was still one, one again. One after one to infinity.
At our own beginning, there were no numbers, not even one. We had no need for numbers--no need to count, no need to know how many. Sufficient unto itself, and for our survival, was each person, each thing, each moment.
Very unlike a divine man would he be, who is unable to count one, two, three, or to distinguish odd and even numbers. PLATO
We knew the day, and the darkness that came after the day, night after day and then day after night. But eventually the time must have come when someone wanted to keep track of yesterday, today, and tomorrow, and count what fills those days: moons and meals and springtimes. And eventually, someone wanted to count what was his--animals, perhaps, or arrows, or seeds, oil, and grain. Perhaps someone wanted to know what was coming--how manydays it would be necessary to wait until the floods would come again, or the moon would disappear and then come back, how long before the baby would be born, or how long before the sun returned from its trip to the edge of the world and the cold passed and the days slowly began to grow longer again.
There is an unexpected story in the history of how we learned to count--from our first recognition of numero uno, the one we mean when we point to ourselves, to the vast numbers we think of when we look at the stars on a moonless night.
Counting is a natural process, almost inevitable, and numbers are organic. They begin with the single line of our bodies, the psychic feel of ourselves. They grow, one by one by one, but they always remain intimately connected to our physical being, from the twoness of our eyes and arms and legs to our ten fingers and--should we need them--our ten toes.
The concept of number is the obvious distinction between beast and man. Thanks to number, the cry becomes a song, noise acquires rhythm, the spring is transformed into a dance, force becomes dynamic, and outlines, figures.
JOSEPH DE MAISTRE
Counting is as natural as numbers. We count each other, and then our children, the things we own, the days we've passed through. Numbers count the things of the world, which must have become less untamable as we numbered its parts and learned to give them names. When we drew an animal on the wall of a cave, we made this one, and then that one--one and two--and we gave the animal its name. When we made designs of the stars over our heads and told their stories to each other, we remembered how many stars there are in Orion's belt or in Cassiopeia's chair. Everywhere, we learned numbers this naturally, and taught ourselves to count, because we always needed to know how many.
Once, we wanted to know how many cattle left in the morning and came home at night, how many seeds it was necessary to savefor next year's planting, or how many days there were from full moon to full moon. Now we need zip codes and Social Security numbers, phone numbers and license plates. Our passports and our houses are numbered, as are our charge accounts and checks and telephones. Numbers have grown away from the simplicity of you and me and the baby. Now, they define us with increasing complexity, and we lose track of how they began, one by one by one--this one, that one, those, and me, you, and the rest of the world.
In another way, perhaps God taught Adam to count when he crafted Eve from Adam's rib and suddenly, where there had been one, lo! there were two. Did Adam and Eve keep track of their children by counting? Did they subtract one from two when Abel disappeared and only Cain was left? Apparently not. But clearly, God could count.
Once we left the garden, learning to count for ourselves was a slow process--it took millennia to learn to answer, in all the varied ways, the basic questions of mathematics: how many? how much? Like magic, numbers became visible as we needed to know them. Along the way, people counted in different ways in different places--by twos, fours, fives, twelves, twenties, sixties, and finally, by tens. Everywhere, numbers became part of civilization, and then came to mean more than just the amount they stood for--they meant good luck and bad, wishes and fairy tales, religion and a way of forecasting the future.
Although we learned numbers at different times and places, we always began with one. Here, we used piles of pebbles as equivalents for numbers, and there, we made notches on sticks or bones, but everywhere, even today, men and women have used their fingers to count. We forget how organic numbers are--how they grew out of our bodies as we learned to hold one finger up to mean one, or point to our eyes for two, and to find higher numbers on our fingers and our toes.
It isn't necessary to count the way we do now in order to answerthe basic question of how many. Little children can say numbers from one to eleventy and beyond, but they have different ways of telling you how many toys they have--this one and that one and the other one, maybe, or just twenty-teen-two.
COUNTING WITHOUT NUMBERS
Do animals count the same way we do? No matter how well we think of ourselves, we aren't alone in grasping the concept of number. Even the lowly rat can learn to press a lever a specific number of times, or take the third right turn (not the second or fourth) in a maze that leads to a reward. Pigeons can learn to peck at a target a specific number of times--thirty-five, not forty, or twenty-three, not nineteen. Chimpanzees will choose a tray that holds seven chocolates over one that holds six. Clever Hans, the famous counting horse, used almost subliminal cues from his trainer and figured out from that--not from the number--how many times to stamp his hoof, but all these animals are, in some way, counting.
Recent research leads us to believe that counting may be simply hard-wired into our brains, whether we're pigeons or people. Monkeys in a study at the Massachusetts Institute of Technology linked computer frames containing a number of dots to frames that were different but contained the same number of dots. Neurons in the monkeys' prefrontal cortex (the part of the brain that makes rapid decisions) rewired themselves when the first frames were shown, and when the same numbers reappeared, the neurons were quickly reactivated--the monkeys recognized the number, even though the frames were different.
But how do cicadas keep track of the years they need underground? They rise through the green grass and into the trees once every seventeen years to make a new generation, and thenreturn through pale roots to the dark underground, to wait and count through all those slow winters before the precise springtime comes and they are ready to rise again, thousands of them all at once, to find their mates.
How do ants find their way home? Or, for that matter, how do they know the way to return to a food source, once they've found it? Researchers trained desert ants to travel from their nest to a food source, and then, to test whether the number of steps they had to take affected their ability to locate the food, glued stiltlike extensions to the legs of some ants and shortened (by cutting) the legs of others. The newly tall ants took the same number of steps they had already learned, and went right past their food. The shortened ants traveled only part of the way to their food goal. According to a 2006 issue of Science, as the ants became used to their new leg length, they adjusted their internal pedometers, in there counting away, and learned how many steps they now needed to take from the nest to the food and back again.
A group of wasps places dead bugs in the cells of developing wasps to be used as food. Different kinds of wasps use different numbers of bugs, but each is consistent, always using five, or ten, or twenty-four. In one group, ten bugs are always placed with female eggs but only five with males, because the female wasps are larger than the males. Somebody is counting.
Perhaps the bugs were arranged in patterns, to be recognized in the same way that we recognize the number of dots on a pair of dice. We don't actually count one-two-three-four, up to six; we look at the pattern and we know. Maybe that's what the wasps do.
Dolphins can recognize strings of as many as eight abstract figures. People, smart as we are, can go up to only six, or at the most, seven. We have to put hyphens in our telephone numbers--one after the area code, and another after what used to be called the exchange--because it's easier to remember a group of three numbers, another group of three, and a group of four, than it isto remember an uninterrupted group of ten. Zip codes stop at five; the added specific code of four more numbers is separated from the zip numbers by a hyphen. Is it counting to remember a phone number? Or is it the memorization of a group of words that happen to mean a number?
We don't usually stop and count anything we're looking at until the group goes higher than four or five. We look at the pattern--a square with something at every corner for four, and with something in the middle for five, or the shape of a triangle with something at each of the angles--and we know the number. We're recognizing the pattern, and we know the pattern means a number. A two-car garage has two doors and holds two cars, and that's as much as most of us need to know about it. But six and seven and eight and, of course, beyond--that's too many to count at a glance. We're not dolphins.
There was a group of Indians in South America without any number words beyond three. But they could recognize groups of things, and knew whether they were complete or not. When they traveled, they were accompanied by their dogs--and while they didn't know how many dogs they had, they knew at a glance if one was missing, and they'd call until the dog returned. Teachers on a class trip work in somewhat the same way. They have a feel for the way the group should look, and if it doesn't look right, then they stop and count. That kind of glance--the pile is smaller than it should be--gives you important information (something is missing), but it doesn't give you a number, it doesn't say how many you have. This kind of counting, the pattern at a glance (there's actually a word for it: subitizing), works only with relatively small numbers. If the teacher has thirty or more kids with her (and no parents, alas), she'll be hard put to know without counting when one is missing.
Up to a point, birds can do the same thing. Take one egg from a nest of several and the bird pays no attention. Take two, and thenest is abandoned. A nineteenth-century astronomer and mathematician, Sir John Lubbock, wrote about a landowner who was bothered by a crow that had chosen to nest in his watchtower. He'd go into the tower to shoo away the bird, and the bird would fly outside and wait until the man had left--and then it would fly back to the nest. After a while, the landowner thought of a way to fool the bird. Two men were sent to the tower with instructions for one to wait inside while the other left; the expectation was that when the bird saw a man leaving, it would feel safe and fly back inside, where the remaining man could deal with it. But the bird was smart enough to wait until they had both left before it returned to the nest. The next day, three men went into the tower and two men left. The clever bird wasn't fooled; it waited patiently again until all three men had left, and then flew happily back inside the tower. Four men entered the tower; three left; the bird waited for the last man before flying back home. At last, five men entered the tower; four left; and this time the bird flew back in again while the fifth man was still there. Conclusion: the bird could count to four, but not to five.
The first principle is that you must not fool yourself--and you are the easiest person to fool.
RICHARD FEYNMAN, COMMENCEMENT ADDRESS AT CALTECH, 1974
The bird was aware of quantity, even if not of number, even if up to only four. This is the beginning of counting. To be aware of the self is the first step toward being aware of the other; and knowing that there are two--me and you--is the first step toward counting the rest of the world. Eventually, we'll know that counting has no end--that it goes on forever, and even then, there can still be one more. In order to do that, to count to forever, we'll have to have the numbers that will make it possible to do so.
We can reach all our numbers, to count to forever, using very few numerals and words. We only have ten digits to work with, no matter what we're counting: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. Eachof those numerals has its own name--one, two, three, and so on, past ten to twelve. After that, for a long time number names are just a combination of the words that have gone before: thirteen is three and ten; twenty is two tens. Ninety-nine is nine tens and nine. When we add one more, we reach a hundred, the first new number name since twelve. The next is thousand, and the next after that--a long way away--is million--so few words and digits for so many numbers!
It's not the voting that's democracy, it's the counting.
TOM STOPPARD, JUMPERS
In exactly the same way, there are only twenty-six letters in the English alphabet, enough to encompass both Shakespeare and the Marx Brothers and everything before and after. A dictionary, said Anatole France, is the universe in alphabetical order. There was only one Shakespeare; there were three principal Marx Brothers; there are millions of words in the dictionary and numbers in the universe. But there are still only twenty-six letters in the English alphabet and a mere ten digits. Those ten numerals are enough to count all the words ever uttered by anybody, and all the letters in all the words. Further, the ten digits are all we need to count every grain of sand, every star, every thing--even every no thing, as the distance of empty space--in the universe.
ONE WAS FIRST
The first number to be thought of, inevitably--because it was the first to be needed--was one, that inward number. One might be enough for a while: one is today, one is now, one is self, one is me, one is first.
As long as one is the only number you need, life is simple. Ifyou have only one of anything, you needn't count it. It's either there or it isn't. You know right away: you see it or you don't. It's not until you add another and another--and another--that keeping track becomes complicated and you need more numbers. Numbers are a sign of plenty.
Even with plenty, though, you can get by with using just one for a while. The notches on ancient bones, the first written countings we know about, are simply lines of one, like notches on a gun stock or scratches made on a jailhouse wall when the sun goes down.
Those lines carefully etched into an animal bone must have come after the very first countings, the ones that we made on our fingers. The index finger points when it's held horizontally; it counts when it's upright. One finger; one number. Most counting systems are based either on five (one hand), ten (two hands), or twenty (fingers and toes). The Latin word for fingers is digiti; our fingers are digits, and numbers are digits as well. In the Middle Ages, digiti was used to mean the single numbers of the decimal system, while articuli ( joints) meant the tens; then digiti came to mean the numerals themselves. When we talk about digital computing, we mean computing by numbers--rather quickly.
There are great advantages to counting with your fingers. With any luck, they're always there. They're clearly visible--other people can see them if you want them to. You can feel what you count. They're portable; wherever you go, they go with you. The problem is, though, that they aren't permanent. You hold up two fingers, and everybody understands that you mean two. When you put your fingers down, two disappears. It leaves no record. You can't walk around for days with two fingers in the air--you'll need them for other things. Unless you have more hands than most of us do, you can't count very high and you can count only one kind of thing at a time--birds or bananas, but not both. (More on finger counting, which can be surprisingly complex, in later chapters.)
There had to be better ways. And there were. Everywhere, always, there was someone who found a better way.
ONE TO ONE
As human beings, we got our start in warm places--Africa, and what we call the Middle East. Fruit and grains grew naturally in our garden, and for a long time we were happy there, nibbling on the things that grew on trees. The years--the aeons--went by while we ate, until we were full of fruit, and so we were fruitful, and we multiplied. We began to wander, and in the new places we found, we hunted and scavenged and made our way, until gradually and slowly, we learned to tame the wildness--and that, in turn, is what tamed us.
We learned to plant seeds, and we stayed to harvest what grew from them. We had farms, and soon, we had neighbors. There were even villages--small groups of farms and people supplying each other's needs. If one family had cows, another might have extra grain, and a third olives for making oil, or grapes for wine. And thus we began to learn things beyond sowing and reaping and tending the animals, because we wanted to trade, and how can you do that--or sell--if you can't count? How can you measure your land if you don't have any kind of numbers? And if you begin to know ways to count--without knowing any numbers--how can you keep track of your countings?
After fingers, the first counting was simple: just one for one. When the cows went out to the field in the morning, the farmer made a pile of pebbles: one pebble for each cow. (Our words calculate and calculus come from the Latin calculus, a pebble, and the Greek khaliks, rock, or limestone.) At the end of the day, the cows came back, and one by one, the farmer removed the pebbles from the pile until all the pebbles were gone. If one pebble was left, onecow was missing. If somebody asked how many cows the farmer had, he could point at the pile of pebbles and say with authority, "That many." We look at an auditorium full of empty seats; the auditorium holds five hundred people. When the seats are filled, we know without counting how many people are there. Prayer beads, used by Muslims, Buddhists, and Catholics, are another kind of one-to-one counting: one bead for each prayer, so it's possible to pray without losing track of how many prayers have been said in the sequence.
The pebbles worked for a beginning, but they were only a bit more useful than fingers: They're slow going, one after one after one; piles of pebbles can tumble over; one pebble looks like another; and it's difficult to keep a permanent record. Anybody could accidentally knock the heap over, pebbles can easily get lost, and a pile of pebbles is hard to move around. Again, something better was needed.
What came next was remarkably permanent: notches in bones. The oldest--a baboon's thigh bone with twenty-nine notches--is thirty-five thousand years old. It was discovered in the Lebombo Mountains of Africa. A wolf bone, found in Europe, is thirty thousand years old. It has fifty-five notches. Our word tally comes from the Latin talea, cut, as a cut twig.
The bones (and wooden sticks) were easier to use than pebbles--a stick could be held in one hand, and as each thing, each sheep, perhaps, came back to the fold, the thumb could slide from one notch to the next. When a new lamb arrived in the springtime, a new notch could be carved to represent it. The sticks could be carried--and they could be split in half, so that a borrower could have a permanent and foolproof record of what he had borrowed. When he made good on his debt, the two halves of the notched stick could be matched--and the record of exactly how much had been borrowed would be clear.
Notched sticks are one of man's first inventions. They cameafter discovering how to use tools for hunting, but before inventing the wheel. And so many millennia later, we still count this way when we need to. Men working in Southern California used to keep track of their days by cutting a line for each day in a block of wood; they used a deeper or thicker line at the end of each week, and a cross at the end of each fortnight. Cowboys made notches in their gun stocks for each buffalo they killed--or for each Indian. During both world wars pilots drew decals on their planes to mark the number of enemy planes they had downed, another kind of tallied one-to-one record. In England, notches on wooden batons stood for varying amounts of pounds sterling. Charles Dickens, in a speech given in 1855, told about
a savage mode of keeping accounts on notched sticks ... introduced into the Court of Exchequer, [where] the accounts were kept, much as Robinson Crusoe kept his calendar on the desert island ... . Official routine inclined to these notched sticks, as if they were pillars of the constitution, and still the Exchequer accounts continued to be kept on certain splints of elm wood called "tallies." In the reign of George III an inquiry was made by some revolutionary spirit, whether pens, ink, and paper, slates and pencils, being in existence, this obstinate adherence to an obsolete custom ought to be continued, and whether a change ought not to be effected. All the red tape in the country grew redder at the bare mention of this bold and original conception and it took till 1826 to get these sticks abolished.
The English Royal Treasury had kept its accounts on tally sticks since the twelfth century. Income, expenditures, taxes--all were notched on tallies. And more: double-notched sticks were issued; half the stick, with one set of notches, was redeemable (assuming it matched--or tallied) for cash through the Treasury,which held the matching piece. Thus, the written (or notched) certificate payable to the bearer after it had agreed with its security: a check. The words stock and dividend also trace back to the Treasury's tally sticks--the stock was the specially marked tally stick held by anyone who lent money to the Bank of England, and thus became a share--or stock--holder. A dividend originally was a tallia dividenda--a "stick to be divided"--redeemable through the Treasury.
The Treasury's Court of the Exchequer didn't take its name from bank checks. Instead, it was named after the cloth that covered the table in the room where local administrators came to settle accounts with the Crown. The cloth was checkered so that counters could be placed in squares that corresponded to various amounts; the final sum was entered both in an account book and on a tally stick--and everyone could understand the process, even if they didn't know how to read or write.
In 1782, it was decided that tally sticks would no longer be issued by the Royal Treasury, but they remained valid until 1826. In 1834, the process of burning the vast numbers of outdated tally sticks began in the furnaces beneath the Houses of Parliament. Unfortunately, the fire was so intense that the Parliament buildings themselves caught fire and went up in flames. In his 1855 speech, Dickens continued his description of the Court of the Exchequer's use of tallies to keep accounts. "It came to pass that [the tallies] were burnt in a stove in the House of Lords. The stove, overgorged with these preposterous sticks, set fire to the panelling; the panelling set fire to the House of Lords; the House of Lords set fire to the House of Commons; the two houses were reduced to ashes; architects were called in to build others; we are now in the second million of the cost thereof; the national pig is not nearly over the stile yet; and the little old woman, Britannia, hasn't got home to-night."
Britain wasn't alone in its use of notched sticks. Tally stickswere used in Germany, in Switzerland, throughout Scandinavia, in Indochina, and in a host of other places. The word for contract in Chinese is written with a character made up of three parts: the character for a tally stick on the left, knife on the right, and under both, another character that means large. Thus, a contract is a big tally stick.
In The Universal History of Numbers, Georges Ifrah writes about a French bakery where, until as recently as the 1970s, tally sticks were used as credit cards of a sort. Two small pieces of wood, called tailles, were notched every time a customer bought a loaf of bread; the baker kept one plank, and the customer took the other home with the bread. At the end of the week, the two planks were matched--the notches had to be equal--and the bill was settled.
AHA!
Tally sticks lasted because they work. But even so, even though they're portable and permanent, they have drawbacks. They probably began before settled farming; but once groups of farms had turned into towns and villages, something more, something better, was needed. Land had to be divided and borders established. Workers had to be paid with supplies of grain or jars of oil. Crops had to be kept track of, so that there would be food to last through the seasons, with enough seed reserved to plant again in the spring. Animals had to be counted and recounted. Taxes and tributes had to be paid. Days had to be numbered so that farmers would know when the moon would be full and thus the days longer for working in the fields, and when the sun would fade in the southern sky as the days grew shorter and the nights cold and long.
Carving into bones was cumbersome; wooden sticks were relatively fragile and flammable. Farmers began instead to fashionclay tokens to serve the same purpose--one-to-one counting, still done without numbers. Tokens were easily moved from one place to another, and they lasted as long as was needed. Most important, they made the next step more obvious: there could be more than a single shape. A round token could be equivalent to one of whatever was being counted, and an oval shape, ten. And then: how much easier to have ten oval tokens instead of a hundred round ones! What a simple but amazing leap forward--no less so because from our point of view it may seem inevitable. The road is rarely clear without a map, and the first counters were traveling in an unknown country of numbers.
There were many moments of genius along the way to where we are now (and there are undoubtedly more to come)--pure Aha! moments that seem so simple when we look back at them, but were so complicated and hidden before somebody thought of them. We learned to make tokens instead of carving bones, and we learned to make simple shapes that represented different amounts. The next leap was considerably larger: tokens could be shaped differently not only to show quantity, but also to show the object being counted--jars of beer or wine could have one shape; loaves of bread another; you could tell by looking (without written words) what it was that was being counted. First the object-shape, and then the number-shape.
No man acquires property without acquiring with it a little arithmetic also.
RALPH WALDO EMERSON, REPRESENTATIVE MEN
It was the Sumerians who took this idea and made it marvelously bold. They lived in the Fertile Crescent, the sweet soil of the land between the Tigris and the Euphrates rivers, in what is now southern Iraq. Their early settlements grew into towns and villages, and then into cities. The largest was Ur; about twenty-five thousand people lived there, with an additional twenty thousand in what we'd now call the metropolitan area. There were other large cities in the FertileCrescent, and inevitably they began to specialize and to trade--timber for gold or silver, barley and corn for oil and wine. In order to trade, they had to ship their products by land or by water, and they had to keep track of what was going and what was coming.
By the second millennium BC, they had the clay tokens. But problems remained. Someone might send twenty-five weights of grain, which would be accompanied by twenty-five tokens. There was no way to stop the shipper, if he was greedy and daring enough, from taking one weight and one token--the receiver would never know what had happened, that he wasn't receiving what he was paying for. The seller, equally unknowing, could also be shortchanged. In order to prevent all this from happening--and keep suspicion at bay--the sender had to know exactly what had been sent; the record had to accompany the goods while they traveled; and the receiver had to know he was getting what he'd paid for.
Clay tokens couldn't do the job--too easy just to take one. They could be put in a pouch--maybe that would work--but then, anybody could open it. Something that would last was needed, to accompany the shipment until it had arrived safely. The solution? Wrap the tokens in clay, and seal the clay-wrapped tokens by baking the whole thing. When the shipment arrived, the buyer could break open the clay package and count the tokens inside. A match made everybody happy.
For a while. Once the packages were broken open, their usefulness was over. All that was left was a pile of tokens. Given that there were three parties involved in each transaction, the sender, the shipper, and the recipient, each had to know that the other two were honest--that the count matched the shipment from beginning to end. There had to be another step.
What if the sender drew a picture on the outside of the clay wrapper showing the number of tokens inside? Then the package could be sealed so that if anybody tried to open it, the receiver would know. More: what if the sender used the same theory thatthe tokens themselves were based on? He could draw one shape to represent grain, or another to stand for oil, and then he could make a second set of symbols to show quantity--one shape, a simple line, to mean the number one, another for ten. Now there would be a picture on the outside that exactly matched the inside, with the sender's seal to validate the package. That would do it! When the seal was broken and the bag was opened, the tokens inside would match the picture on the outside, and the shipment would be safe.
But Aha! After all that, having come this far, why bother with the inside, with all those pesky little tokens? Everything that everybody needed to know was clearly depicted on the outside of the clay envelope. Forget the tokens! When the shipment arrived, the shipper could double-check to see that the figures on the tablet matched the shipment, and the purchaser could clearly and easily tell whether or not he'd been cheated, or if was dealing with honest men. The wrapper was no longer a wrapper. It was the thing itself. It was a clay tablet. It was an invoice. And the pictures were written on it. It was a count, it was a record, and it was writing. Counting came before writing. Counting was writing. We counted because we needed to, and then we learned to write down what we had counted. And from there, we flew down the years to poetry and libraries and shopping lists and constitutions. One, two, and three. And more to come.
ANOTHER NOTCH
For such an important number, the beginning of everything, one is quite plain. It looks simple. It has no subtlety; it just stands there, straight and tall, entire and unadorned. It has no curls or loops, no corners to speak of, nothing soft, no womanly curves, just one firm line, top to bottom, solid. At the most, it has half an arrow on the top--or, if you prefer, a little hat, tilted to one sidein a somewhat jaunty attempt to look debonair. It sometimes has a tiny platform on its bottom to stand upon, but it remains modest and single, somewhat stubborn, unbreakable as it is except into fractions. Its very solidity and stubbornness make it quite safe, but there's no connection there. I think of one as lonely and a bit dull--or at least rather bored. Alone and lonely, after all, both have one in them; and only is just one, and none is worse, not even one.
One is one and all alone and ever more shall be so.
"GREEN GROW THE RUSHES," ENGLISH FOLK SONG
One can indeed accomplish so much (one plus one forever), but it's left with so little. It must be filled with longing (it's a bachelor--definitely phallic), yearning for another, as Adam did, all alone in Eden. There are solitary joys, and some of them are splendid, but so much of ourselves is built around the idea of two: eyes, ears, hands--two is almost as basic to our natures as one. More so, if the need to share is recognized. "Look!" we want to say when we see something unusual. Or "Did you hear that?" One is the beginning, the essence, one is what everything is made of, and what we must always return to, but there's no getting around it--one is alone, and can be lonely. One is myself, alone at first and alone at last.
But whatever comes next, one was here first. And in the other direction, one is all that separates us from nothing, the void, extinction--zero, though zero arrived a long time after one had left nothing behind. One was first.
One is distinction and difference. One is apart; one is known. One is a place, a thing, a dot, a point. There is no counting without one, though one doesn't need to be counted--it's simply there. But then, there is no anything without one.
One is the integer, the whole number, and one is integrity. The simplicity and singularity of one are deceptive, for one iseverything. It embraces all, and encompasses infinity. It can stand a little loneliness.
The first number word in English, one, comes from the Latin word unus. Unus is the root for words like union, unity, unison, unique (one of a kind), and unanimous (of one mind). It's also the source, indirectly, through Anglo-Saxon, for a and an, both of which mean one. An apple is one apple; a banana is one banana. To be all of one piece is to be whole, which gives us words like holy, wholesome--even hale and healthy. Atone is one too, to be at one again, and so is once, when it happened first.
The earlier Latin for one was oinos--strangely like an onion (which could be defined as one sharply tasty sphere encompassing many layers). The Indo-European prototype was, variably, oi-no, oi-ko, or oi-wo. In Sanskrit, the word is eka; in German, ein; in Russian, odin; in Irish, oin.
Old Egyptian hieroglyphs through all the kingdoms and dynasties show the number one as the familiar straight line. Egyptian carvers, in the later years, apparently liked puzzles and word games, and in some of their inscriptions used pictures to show numbers. One was sometimes depicted as a round circle--the sun, because there's only one sun, even if it does keep reappearing--or it could be shown as a small upward curved line with the top of a circle peeping over the top, representing the moon, for the same reason.
In something of the same spirit--though not as a puzzle--Indian mathematicians used a variety of poetic concepts to express numbers. The numerals we know trace back to ancient India; in the eleventh century, a Persian astronomer wrote about the numbers used in Indian astronomical tables, and said that when it was difficult to write the word for a number in a certain place in the tables, astronomers could choose from "amongst its sisters." Quoting the great seventh-century Indian astronomer and mathematician Brahmagupta, he went on, "If you want to writeone, express it through a word which denotes something unique, like the Earth or the Moon." Among the other sister words for the number one were "the Ancestor," referring to Brahma, considered the creator of the universe, "Beginning," and "Body."
THE ROMANS AND THE GREEKS, FOR A CHANGE
The Latin words printed on American coins tell us that one means coming together in unison: e pluribus unum--out of many, one. The ancient Greeks--the Pythagoreans--saw one differently, in the opposite way: out of one, many.
Numbers have always been a way of explaining the world, whether in Sumerian tokens and tablets, Einsteinian formulas, or the all-encompassing view of the i. Pythagoras was born in about 580 BC on the island of Samos, traveled as a youth to Egypt and Babylon, and eventually founded a school in Magna Graecia (southern Italy) which became the basis of the secretive Pythagorean Brotherhood.
Three hundred rich and powerful young men attended the school. They were divided into two groups: the outer circle, the akousmatikoi (those who hear--related to our word acoustics) learned the group's rules of conduct. Once these were mastered, its members could progress to the inner circle, called the mathematikoi. (Pythagoras is credited with having coined the word mathematics, to mean "that which is learned," as well as the word philosophy, "the love of wisdom.") They studied the most secret and difficult of Pythagoras's truths--in the areas of geometry, astronomy, number theory, and music.
Music was basic to their studies, for Pythagoras believed that the universe sings--that there is a literal music in the skies, formed by the motions of the planets and made up of tones based on numerical frequencies and ratios. He believed numbers enable us tounderstand that music's harmonies--but even beyond that, he believed numbers existed before there was physical reality, even that numbers create and are physical reality, the building blocks of the universe, the cause of everything that exists. Only through number and form, the Pythagoreans held, could man grasp the nature of the universe. Everything that is can be numbered.
The odd and the even are elements of number, and of these the one is infinite and the other finite, and unity is the product of both of them, for it is both odd and even, and number arises from unity, and the whole heaven, as has been said, is number.
ARISTOTLE
All of this began when Pythagoras discovered the relationship between the length of a string and the sound it makes when it's plucked; he realized that if the string is shortened to half of its original length, the tone it makes is an octave higher than the original note, a sound he described as pleasant and "in harmony." He then worked out the string lengths necessary for other harmonies--at three-quarters of its original length, the string produces a tone which we call a fourth; at two-thirds of its length, the tone is a fifth. While we know that the tones change because the vibrations of the string change at its different lengths, Pythagoras believed instead that the harmonies depended on the ratios between numbers, and that, eventually, all things depend upon numbers. For him, the heavens were a musical scale.
Music is the arithmetic of sounds as optics is the geometry of light.
CLAUDE DEBUSSY
 
 
I myself figured out the peculiar form of mathematics and harmonies that was strange to all the world but me.
JELLY ROLL MORTON
(In a way, Einstein believed something similar. He once said that Beethoven created the music he wrote, but Mozart's music "was so pure that it seemed to have been ever-present in the universe, waiting to be discovered by the master." Einsteinsaw physics in that way--that beyond observation and theory was the music of the spheres, a "pre-established harmony," waiting to be discovered.)
Pythagoras believed in the migration of the soul after death and developed a ceremony to purify the soul in readiness for its journeys. Because the souls of friends might return as animals, vegetarianism was preferred to meat eating. But the eating of beans was banned. Aristotle said that among the reasons for the bean taboo was the possibility that beans may have arisen simultaneously with humans in the moment of the universe's creation--and also that beans resemble genitals.
The Brotherhood was religious as well as mystical, and practical as well as either. It developed principles that influenced Plato and Aristotle, and much of Western rational philosophy. The first mathematical proofs were the Pythagoreans'; they developed mathematical theory still in use today--you learned about right triangles in high school geometry. (The sum of the squares of the lengths of the two shorter sides of a right triangle is equal to the square of the hypotenuse, which is the longest side.)
But beyond the hypotenuse, much of the Pythagorean tradition is in the realm of mystical wisdom, rather than science and scholarship, mathematics and geometry, as we know them today. A good example is the Pythagorean belief that odd numbers (one, three, five, seven, nine ...) are indissoluble, therefore masculine and celestial in nature, and that even numbers (two, four, six, eight, ten ...) are soluble, therefore feminine, ephemeral, and earthly. This male-female duality existed for the Pythagoreans beyond the male-female division of numbers. They divided reality into two parts: the mind and the spirit were aligned with the realm of the gods, which was male; the body and matter were the realm of the earth, which was female. Overall, numbers belonged with the male; thinking about numbers and the tasks of mathematics were masculine work, associated with the gods andwith transcendence from all that was material. "It is here," Margaret Wertheim wrote in The New York Times in 2006, "that we begin to see the seeds of modern physics." The first universities were founded to educate the clergy. Women couldn't be priests, so neither could they be university students, and physics departments, well into the twentieth century, were often the last to admit women as students or as professors.
(Meanwhile, on the other side of the world, the Chinese believed that odd numbers meant white, day, heat, sun, and fire, and even numbers the opposite: black, night, cold, water, earth. The Chinese yin and yang represent odd and even numbers. The yang is masculine, and represents the sun, day, summer, light, and openess. The yin is feminine--the moon, night, winter, shade, secrecy. They alternate, one after the other, together and touching, one and two forever.)
For the Pythagoreans, individual numbers not only had abstract qualities like masculinity and earthliness, but also could be identified with human attributes. One, because it's unchangeable, they linked to reason; two, with its pairs and opposites, to opinion or polarity. Harmony was a property of three. Four stood for justice because it's the product of equals (two plus two) and because squares, with their four even sides, are perfect figures; four also represented space and matter. Five represented marriage because it's the first union of a feminine number with a masculine number (two plus three equals five)--and so on through the numbers to a perfect Pythagorean ten.
The trouble with one, for the Pythagoreans, was that for all its unchangeable reasonability, they didn't believe one to be a number at all. Numbers were totalities composed of separate units--something plus something--but one, they thought, is a totality complete unto itself. (They had no truck with fractions.) For them, there was one, and there was more than one. One is "that which is"; it's a statement of existence: one is. One doesn't have separateunits; it is a unit, the unity, the opposite of the idea of many, the opposite of plural. One is pure and strong, and they believed it to be the single indivisible thing from which numbers arose--not the other numbers, because one was not a number, but the numbers. Numbers were an idea sort of like the offspring of the first man, the one, who was Adam. One is the father number. (And two--like Eve--is the mother number. But we're not at two yet.)
That concept of one, as the source of numbers without being a number itself, lasted through the Middle Ages. A twelfth-century manuscript, the Salem Codex, said, "Every number can be doubled and halved, except for unity; this can, it is true be doubled, but not halved--wherein lyeth concealed a great Mysterium [God]." A sixteenth-century German mathematician agreed: "1 is no number, but it is a generatrix, beginning, and foundation of all the other numbers." In Number Words and Number Symbols, Karl Menninger quotes Michael Stevin as the first mathematician to say--in 1585--that one is a number. He proved it by saying that if you subtract one from three, you are left with two; if one were not a number, subtracting one from three would leave three untouched.
Thousands of years after the ancient Greeks, Freud found one to be a perfect phallic symbol. It is upright and solid, as neat a phallic symbol as a stalk of asparagus--better, probably, because it isn't green. Whether it's because of its perfectly pricklike shape or whether it is some kind of numerical and linguistic testimony to the nature of the male, there are languages in which the word for one and the word for man--and more, even just the word for penis--are the same: man, penis, one--it all comes to the same thing.
There are also languages--other languages--in which the word for God is the same as the word for one, and two is the same as the word for sin--because two is the first step away from one, and therefore is a step away from God. In other languages, early Hebrew and Arabic among them, numbers began with two--asthey did for the Pythagoreans, but here it was because one was the number for God (as "There is one God," or "God is One"), and was reserved for God alone.
FIRST AND ONCE AND ALWAYS
Mathematically, one is the first odd number. But then, one is the first everything--one almost defines first. You can't have anything first without having one.
First is a word that does hard work as an adjective as well as a noun. It covers everything from the big toe on your foot to the opening lines of a poem to the base immediately to the right of home plate in baseball. It deals with time--it can be the earliest thing--as well as your car's lowest gear; it comes before all others in rank or occurrence; it's the beginning and it's the winning number. The president's wife is the First Lady; someday, perhaps the president's husband will be the First Gentleman, if not the First Man. On a first-name basis is the beginning of friendship. First place is best.
The last thing one knows when writing a book is what to put first.
BLAISE PASCAL, PENSEES
First is the ordinal word for one, which is a cardinal word. In other words, one is a number in a series--one, two, three, eventually telling you how many there are; first--or second or third--tells you the number's position in the series.
First and second are words that have nothing obvious to do with one and two, though clearly third, fourth, fifth--and so on--are derived from three, four, five ... . The same thing is true in other languages--all the way back to Greek and Latin and before. First is related to words meaning "before," or "in front"--the Indo-European pro, "before," and theSanskirt puras, "in front," eventually evolved (with the fr sound substituting for the pr sound) into foremost, and then to first. (In German, first is erst, which originally meant "early morning.") Second is related to words meaning "the other," as the Latin alter, and the Indo-European root anteros, meaning "the other of two" or "the following," and the Latin sequi, "to follow," and secundus, "the next after the first."
In many places, counting went only from one to two--one was singular, two plural, and beyond two was a vast multitude, the idea of many. In the same way, it may be that it took us just that long to get the connection between the cardinal count of one and the ordinal place of first, and then two and second. Second was disconnected from two; rather, it was simply the next. By the time we reached three and third, we'd not only understood the number after two, we had also begun to understand the relationship; third wasn't just one after one after one. It was one in a series--the third, in fact. We're not that different from a tribe on Papua New Guinea. In their language, Ponam, the only words that show placement are first, middle, and last--nothing else matters, even though Ponam has many numbers to count with.
The advantage of a bad memory is that one enjoys several times the same good things for the first time.
FRIEDRICH NIETZSCHE
Primary and secondary, again from the Latin primus and secundus, "first" and "second," are followed by tertiary (third), quaternary, quinary, senary, septenary, octonary, nonary, denary, duodenary--which takes us to twelfth--and then there's a leap to vigenary, twentieth. There was a tiny leap from tenth to twelfth--the eleventh place has no single word of its own, alas. In the same way, once, twice, and thrice are followed by a total blank--there are no single words for four times, five times, or any other number of times; apparently, says the Oxford English Dictionary, a usually reliable source, our language has never needed any numerical wordsafter thrice. Yet. Note that quince is a fruit, and has nothing to do with how many times you've been able to find one growing on a tree. Its name derives, in rather roundabout ways, through a variety of words basically meaning a kind of apple--though if you cut an apple in half across its middle, you'll find a five-pointed star.
Another quirk in this category of numerical word oddities is the pronunciation of one--and for that matter, once. Words derived from one, like only, alone, and atone, give the o a long sound--what Mrs. Quinn, my fifth-grade teacher, would have said was a letter saying its own name. O as in go, toe, row, over, and oh. Apparently, the way we pronounce one--as wun instead of w-oe-n--can be traced back to western England and Wales in the Middle Ages, when the vowel sound began a series of changes--from o as in own to o as in boot to o as in took and then o as in one. The final sound stuck and then it spread. Other vowel sounds changed too in those places (oats, for example, went from the long o sound of oh-ts to the sound of the u in cuts) but only the pronunciation of one and once stayed around to sound as we use it today.
"Can you do addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?"
"I don't know," said Alice, "I lost count."
LEWIS CARROLL, THROUGH THE LOOKING-GLASS
The pronunciation of one, then, is an oddity, and one is an odd number, at least mathematically. Odd numbers can't be divided easily into two equal piles--there's always something left over. They're solid and smug. Like Popeye, they seem to say, "I yam what I yam." And they've always just eaten a can of spinach.
At heart, though, there's nothing odd about one. It is a number; it's a splendid number. It's part of the endless series that reaches to the stars but begins, always, with the number one.
If you have one number, you have every number, because every number has to begin with one, and because no matter whatnumber you have, no matter how big a number it is, you can always make it larger by adding one more. Isn't that what infinity is? The last number and then one more ... and one more ... and one more ... forever.
All for one, one for all.
ALEXANDRE DUMAS THE ELDER, THE THREE MUSKETEERS
One is first because, just as the Pythagoreans said, all the other numbers came from one. Without one, there is nothing. With one, there is everything. Everything begins with one. One makes it possible to go on forever.
One is all.
PERFECT FIGURES. Copyright © 2007 by Bunny Crumpacker. All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles or reviews. For information, address St. Martin's Press, 175 Fifth Avenue, New York, N.Y. 10010.